*– Отец, а пространство бесконечно?**– Конечно, сын мой.*

Investigating the laws of the Universe, scientists usually come to their
conclusions using either induction (*from
particular to general*) or deduction (*from
general to particular*). The first methodology is invalid since it is all based
on the principle of extrapolation which can be confirmed only by intuition. The
second methodology is also quite arguable because it relies on some general
laws which are clearly obtained through induction. In some sense, all the
fundamental knowledge we have is derived by means of induction and therefore is
unreasonable.

The area of knowledge which seems to be less doubtful is mathematics and logic particularly. In fact, logic is an essential part of any scientific method: both inductive and deductive approaches rely on it.

We can subdivide mathematics in applied math and *pure* math. Applied math is being developed to *explain* and *predict* reality.
These two major functions of applied math are used in physics, computer
science, economics, sociology and architecture. Pure math is more theoretical, however
some of its chapters can be applied. Often pure math is done as an art, and great
efforts are spent on things which are inapplicable to real life (for example,
on studying flexible polyhedra). Applied math requires some purely mathematical
results to proceed while pure math is completely independent.

How does applied math work? Dealing with some real-life problem,
scientists begin with modelling reality. To make the model realistic enough, they
make several assumptions based on real observations. Then from these inputs some
conclusions are drawn by means of pure math. Finally, acquired results are
interpreted or translated from the mathematical language of the model to the
language of real life. But can we come to any *true* conclusions this way?

The problem is that models usually are irrelevant due to incompleteness of assumptions. Furthermore, these assumptions are always obtained through induction which validity is questionable. Therefore, according to the principles of logic, all interpretations of the model’s output are inappropriate since the model itself is wrong. The only reason models are useful is that some of them are of enough explanatory (or descriptive) power. They allow us to generalize our previous experience saying that it formed a certain pattern, but do not give us any confidence that this pattern will be hold further on. So, applied math is good at explanation but not at prediction.

Now what is pure math? It is the only piece of knowledge which is whole
derived through deduction. It starts with definitions and axioms *based on nothing*. Then axioms give
foundation to theorems and “practical” methods which help in solving purely
mathematical problems. The key difference from the applied math is that there
is no apparent connection with real life. Pure math is self-sufficient and the
only way how it is linked with reality is that it is used in applied math as a
tool. It allows scientists to derive conclusions which are all wrong from the epistemological
point of view.

Pure math does not explain reality (if not being combined with applied
math) and clearly does not predict it. Nevertheless, it seems to be the only *truth* we know. This great benefit is
more than compensated by the abstractness which at the same time makes pure
math *truth*. In fact, we have no
guarantee that our mathematical reasoning is correct, and this is why it is
better to say pure math is the only area of our knowledge that *can be truth*. As for other fields, the
induction problem leaves just a miserable chance that we are *right*.

A good example of pure math is linear algebra. It operates with
completely abstract notions and objects (such as *n*-dimensional linear vector space) which can model multiple
real-life cases. For instance, linear algebra can be beautifully applied in
physics. In combination with *empiric*
Newtonian or Einsteinian laws it allows astronomers to calculate planets’
orbits and *predict* their movement. This
does not mean that we know how planets revolve: there are many unaccounted factors
in scientists’ models while all the accounted ones easily can be wrong due to
the induction problem. However, we can be absolutely sure that *the determinant of the product of two matrices
is equal to the product of their determinants*. That’s good news, isn’t it?